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2017 | Supplement | Chapter

Long Time Dynamics and Coherent States in Nonlinear Wave Equations

Author : E. Kirr

Published in: Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Publisher: Springer New York

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Abstract

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.

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R. A. Adams, Sobolev Spaces. Academic Press, New York, 1975.MATH

A. Ambrosetti, M. Badiale, S. Cingolani, “Semiclassical states of nonlinear Schrödinger equations with bounded potentials”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7 (1996), no. 3, 155–160.

W.H. Aschbacher, J. Fröhlich, G.M. Graf, K. Schnee, and M. Troyer, “Symmetry breaking regime in the nonlinear hartree equation”, J. Math. Phys. 43, 3879–3891 (2002).MathSciNetCrossRefMATH

D. Bambusi, S. Cuccagna, “On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential”, Amer. J. Math. 133 (2011), no. 5, 1421–1468.MathSciNetCrossRefMATH

V. Benci, D. Fortunato, Variational methods in nonlinear field equations. Solitary waves, hylomorphic solitons and vortices. Springer Monographs in Mathematics. Springer Cham Heidelberg New York Dordrecht London, 2014.

H. Berestycki, P.-L. Lion, “Nonlinear scalar field equations”, Arch. Ration. Mech. Anal. 82 (1983) 313–375.

N. Boussaid, E. Kirr, “Asymptotic stability of ground states in Dirac equation”, in preparation.

B. Buffoni, J. Toland, Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2003.

V. S. Buslaev, G. S. Perelman, “Scattering for the nonlinear Schrödinger equation: states that are close to a soliton”. St. Petersburg Math. J. 4 (1993), no. 6, 1111–1142.

10.

V. S. Buslaev, G. S. Perelman, “On the stability of solitary waves for nonlinear Schrödinger equations”. Nonlinear evolution equations, 75–98, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995.

11.

V. S. Buslaev, C. Sulem, “On asymptotic stability of solitary waves for nonlinear Schrödinger equations”. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 3, 419–475.MathSciNetCrossRefMATH

12.

T. Cazenave, Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics (New York University Courant Institute of Mathematical Sciences, New York, 2003).

13.

S. Cuccagna, “Stabilization of solutions to nonlinear Schrödinger equations”, Comm. Pure Appl. Math. 54 (2001), 1110–1145.MathSciNetCrossRefMATH

14.

S. Cuccagna, T. Mizumachi, “On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations”, Comm. Math. Phys. 284 (2008), no. 1, 51–77.MathSciNetCrossRefMATH

15.

A. Floer and A. Weinstein, “Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential”, J. Funct. Anal. 69, 397–408 (1986).MathSciNetCrossRefMATH

16.

M. Golubitsky, I. Stewart, D. G. Schaeffer, Singularities and groups in bifurcation theory, Vol. II. Applied Mathematical Sciences, 69, Springer-Verlag, New York, 1988.

17.

M. Grillakis, “Linearized instability for nonlinear Schrödinger and Klein–Gordon equations”, Comm. Pure Appl. Math. 41, 747–774 (1988).MathSciNetCrossRefMATH

18.

M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry. I,”, J. Funct. Anal. 74 (1987), no. 1, 160–197.MathSciNetCrossRefMATH

19.

M. Grillakis, J. Shatah, W. Strauss, “Stability theory of solitary waves in the presence of symmetry. II”, J. Funct. Anal. 94 (1990), no. 2, 308–348.MathSciNetCrossRefMATH

20.

Y. Guo, R. Seiringer, “On the mass concentration for Bose-Einstein condensates with attractive interactions.” Lett. Math. Phys. 104 (2014), no. 2, 141–156.MathSciNetCrossRefMATH

21.

S. Gustafson, K. Nakanishi, T.-P. Tsai, “Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves”. Int. Math. Res. Not. 2004, no. 66, 3559–3584.

22.

R. K. Jackson, communication at SIAM Conference on Nonlinear Waves, Philadelphia, Aug. 2010.

23.

R. K. Jackson, M. I. Weinstein, “Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation.” J. Statist. Phys. 116 (2004), no. 1–4, 881–905.MathSciNetCrossRefMATH

24.

H. Jeanjean, M. Lucia and C. Stuart, “Branches of solutions to semilinear elliptic equations on \({R}^{N}\)”, Math. Z. 230, 79–105 (1999).MathSciNetCrossRefMATH

25.

H. Jeanjean, M. Lucia and C. Stuart, “ The branche of positive solutions to a semilinear elliptic equation on \({R}^{N}\)”, Rend. Sem. Mat. Univ. Padova, 101, 229–262 (1999).MathSciNetMATH

26.

P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, “Vortices in Bose-Einstein Condensates: (Super)fluids with a twist”, SIAM Dynamical Systems Magazine, October, 2011.MATH

27.

E.W. Kirr, P.G. Kevrekidis, E. Shlizerman, and M.I. Weinstein, “Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross–Pitaevskii equations”, SIAM J. Math. Anal. 40, 56–604 (2008).CrossRefMATH

28.

E. Kirr, P.G. Kevrekidis, D. Pelinovsky, “Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials”, Commun. Math. Phys. 308 (2011), 795–844CrossRefMATH

29.

E. Kirr, A. Zarnescu, On the asymptotic stability of bound states in 2D cubic Schrödinger equation Comm. Math. Phys. 272 (2007), no. 2, 443–468.MathSciNetCrossRefMATH

30.

E. Kirr, A. Zarnescu, Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases, J. Differential Equations 247 (2009), no. 3, 710–735.MathSciNetCrossRefMATH

31.

E. Kirr, A. Zarnescu, Asymptotic stability of large ground states in nonlinear Schrödinger equation, in preparation.

32.

E. Kirr and Ö. Mızrak, “Asymptotic stability of ground states in 3d nonlinear Schrödinger equation including subcritical cases”, J. Funct. Anal. 257, 3691–3747 (2009).MathSciNetCrossRefMATH

33.

E. Kirr and Ö Mızrak, “ On the stability of ground states in 4D and 5D nonlinear Schrödinger equation including subcritical cases” submitted to Int. Math. Res. Not. available online at: http://arxiv.org/abs/0906.3732

34.

E. Kirr, P.G. Keverekidis and V. Natarajan, “Bifurcations of large ground states in one dimensional nonlinear Schrödinger equation”, in preparation.

35.

E. Kirr and V. Natarajan, “The global bifurcation picture for coherent states in nonlinear Schrödinger equation”, in preparation.

36.

P.-L. Lions, “The concentration-compactness principle in the calculus of Variations. The locally compact case. I.” Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 109–145.MathSciNetCrossRefMATH

37.

P.-L. Lions, “The concentration-compactness principle in the calculus of Variations. The locally compact case. II.” Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 223–283.MathSciNetCrossRefMATH

38.

J.L. Marzuola and M.I. Weinstein, “Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross–Pitaevskii equations”, DCDS-A, to be published (2010).MATH

39.

T. Mizumachi, “Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential”, J. Math. Kyoto Univ. 48 (2008), 471–497.MathSciNetCrossRefMATH

40.

T. Mizumachi, “Asymptotic stability of small solitons for 2D Nonlinear Schrödinger equations with potential”, J. Math. Kyoto Univ. 47 (2007), no. 3, 599–620.MathSciNetCrossRefMATH

41.

L. Nirenberg, Topics in nonlinear functional analysis, Courant Lecture Notes 6 (New York, 2001).

42.

E. S. Noussair, S. Yan, “On positive multipeak solutions of a nonlinear elliptic problem.” J. London Math. Soc. (2) 62 (2000), no. 1, 213–227.

43.

Y.-G. Oh, “On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential”, Comm. Math. Phys. 131 (1990), no. 2, 223–253.MathSciNetCrossRefMATH

44.

C.A. Pillet, C.E. Wayne, “Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations”, J. Diff. Eqs. 141, 310–326 (1997).MathSciNetCrossRefMATH

45.

P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,” J. Functional Anal. 7 (1971), 487–513.MathSciNetCrossRefMATH

46.

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Analysis of Operators. Volume IV. Academic Press San Diego New York Boston London Sydney Tokyo Toronto, 1972.

47.

H.A. Rose and M.I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential, Physica D 30, 207–218 (1988).MathSciNetCrossRefMATH

48.

I. M. Sigal, G. Zhou, “Asymptotic stability of nonlinear Schrödinger equations with potential”. Rev. Math. Phys. 17 (2005), no. 10, 1143–1207.MathSciNetCrossRefMATH

49.

A. Soffer and M.I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations”, Comm. Math. Phys. 133, 119–146 (1990).MathSciNetCrossRefMATH

50.

A. Soffer and M.I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data” J. Diff. Eqs. 98, 376–390 (1992).MathSciNetCrossRefMATH

51.

A. Soffer, M. I. Weinstein, Selection of the ground state for nonlinear Schroedinger equations, Rev. Math. Phys. 16 (2004), no. 8, 977–1071.MathSciNetCrossRefMATH

52.

T.-P. Tsai, H.-T. Yau, Horng-Tzer Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not. 31 (2002), 1629–1673.CrossRefMATH

53.

T.-P. Tsai, H.-T. Yau, Stable directions for excited states of nonlinear Schrödinger equations. Comm. Partial Differential Equations 27 (2002), no. 11–12, 2363–2402.MathSciNetCrossRefMATH

54.

T.-P. Tsai, H.-T. Yau, Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data. Adv. Theor. Math. Phys. 6 (2002), no. 1, 107–139.MathSciNetCrossRefMATH

55.

M.I. Weinstein, “Lyapunov stability of ground states of nonlinear dispersive evolution equations”, Comm. Pure Appl. Math. 39, 51–68 (1986).MathSciNetCrossRefMATH

56.

M.I. Weinstein, “Localized States and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii Equation”, Frontiers of Applied Dynamical Systems: Reviews and Tutorials, vol. 3 (2015), 41–79.CrossRef

57.

G. Zhou, “Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations”, J. Math. Phys. 48 (2007), no. 5, 053509–053532.MathSciNetCrossRefMATH

Title: Long Time Dynamics and Coherent States in Nonlinear Wave Equations
Author: E. Kirr
Publisher: Springer New York
Book: Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science
Print ISBN: 978-1-4939-6968-5

Electronic ISBN: 978-1-4939-6969-2

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-1-4939-6969-2_3

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